Cot2x - tan 78 = ( sec x sec 78 )/2
Here's another solution
I showed this to Rod and he corrected an error in the final steps, The full solution is as follows.
I let y = 78 degrees in the expression above and simplified it from there.
cot(2x) – tan(y) = ( sec(x).sec(y) )/2
cos(2x)/sin(2x) – sin(y)/cos(y) = 1/(2cos(x).cos(y))
{cos(2x).cos(y) – sin(2x).sin(y)}/{sin(2x).cos(y)} = 1/(2cos(x).cos(y))
{cos(2x+y)}/{sin(2x).cos(y)} = 1/(2cos(x).cos(y))
{cos(2x+y)}/{sin(2x)} = 1/(2cos(x))
cos(2x+y) = sin(2x)/(2cos(x))
cos(2x+y) = 2sin(x).cos(x)/(2cos(x))
cos(2x+y) = sin(x)
cos(2x+y) = cos(pi/2-x)
equating the arguments of the two cosines,
2x + y = pi/2 – x + 2n.pi
3x = pi/2 + 2n.pi – y
3x = 90 + 360n – 78
3x = 12 + 360n
x = 4 + 120n
x = 4 + 0, 120, 240, 360, …